Ответы
Ответ дал:
1
Найдём сначала точки пересечения гиперболы ху=6 и прямой х+у=7 .
![\left \{ {{xy=6} \atop {x+y=7}} \right. \; \left \{ {{x(7-x)=6} \atop {y=7-x}} \right. \; \; \left \{ {{x^2-7x+6=0} \atop {y=7-x}} \right. \; \left \{ {{x_1=1\; ,\; x_2=6} \atop {y_1=6\; ,\; y_2=1}} \right. \left \{ {{xy=6} \atop {x+y=7}} \right. \; \left \{ {{x(7-x)=6} \atop {y=7-x}} \right. \; \; \left \{ {{x^2-7x+6=0} \atop {y=7-x}} \right. \; \left \{ {{x_1=1\; ,\; x_2=6} \atop {y_1=6\; ,\; y_2=1}} \right.](https://tex.z-dn.net/?f=+%5Cleft+%5C%7B+%7B%7Bxy%3D6%7D+%5Catop+%7Bx%2By%3D7%7D%7D+%5Cright.+%5C%3B++%5Cleft+%5C%7B+%7B%7Bx%287-x%29%3D6%7D+%5Catop+%7By%3D7-x%7D%7D+%5Cright.+%5C%3B+%5C%3B+%5Cleft+%5C%7B+%7B%7Bx%5E2-7x%2B6%3D0%7D+%5Catop+%7By%3D7-x%7D%7D+%5Cright.+%5C%3B++%5Cleft+%5C%7B+%7B%7Bx_1%3D1%5C%3B+%2C%5C%3B+x_2%3D6%7D+%5Catop+%7By_1%3D6%5C%3B+%2C%5C%3B+y_2%3D1%7D%7D+%5Cright.+)
![\iint \limits _{D}\, x\cdot dx\, dy= \int\limits^6_1 \, x\, dx \int\limits^{7-x}_{\frac{6}{x}} \, dy = \int\limits^6_1 \, x\, dx \Big (y\Big |_{\frac{6}{x}}^{7-x}\Big )=\\\\= \int\limits^6_1 \; {x\, (7-x-\frac{6}{x})} \, dx = \int\limits^6_1 \, (7x-x^2-6) \, dx =\\\\=( \frac{7x^2}{2} -\frac{x^3}{3}-6x)\Big |_1^6= (\frac{252}{2} - \frac{216}{3}-36 )-( \frac{7}{2} -\frac{1}{3} -6)=\\\\=\frac{245}{2} - \frac{215}{3} -30= \frac{125}{6} =20\frac{5}{6} \iint \limits _{D}\, x\cdot dx\, dy= \int\limits^6_1 \, x\, dx \int\limits^{7-x}_{\frac{6}{x}} \, dy = \int\limits^6_1 \, x\, dx \Big (y\Big |_{\frac{6}{x}}^{7-x}\Big )=\\\\= \int\limits^6_1 \; {x\, (7-x-\frac{6}{x})} \, dx = \int\limits^6_1 \, (7x-x^2-6) \, dx =\\\\=( \frac{7x^2}{2} -\frac{x^3}{3}-6x)\Big |_1^6= (\frac{252}{2} - \frac{216}{3}-36 )-( \frac{7}{2} -\frac{1}{3} -6)=\\\\=\frac{245}{2} - \frac{215}{3} -30= \frac{125}{6} =20\frac{5}{6}](https://tex.z-dn.net/?f=%5Ciint+%5Climits+_%7BD%7D%5C%2C+x%5Ccdot+dx%5C%2C+dy%3D+%5Cint%5Climits%5E6_1+%5C%2C+x%5C%2C+dx++%5Cint%5Climits%5E%7B7-x%7D_%7B%5Cfrac%7B6%7D%7Bx%7D%7D+%5C%2C+dy+%3D+%5Cint%5Climits%5E6_1+%5C%2C+x%5C%2C+dx+%5CBig+%28y%5CBig+%7C_%7B%5Cfrac%7B6%7D%7Bx%7D%7D%5E%7B7-x%7D%5CBig+%29%3D%5C%5C%5C%5C%3D+%5Cint%5Climits%5E6_1+%5C%3B+%7Bx%5C%2C+%287-x-%5Cfrac%7B6%7D%7Bx%7D%29%7D+%5C%2C+dx+%3D+%5Cint%5Climits%5E6_1+%5C%2C+%287x-x%5E2-6%29+%5C%2C+dx+%3D%5C%5C%5C%5C%3D%28+%5Cfrac%7B7x%5E2%7D%7B2%7D+-%5Cfrac%7Bx%5E3%7D%7B3%7D-6x%29%5CBig+%7C_1%5E6%3D+%28%5Cfrac%7B252%7D%7B2%7D+-+%5Cfrac%7B216%7D%7B3%7D-36+%29-%28+%5Cfrac%7B7%7D%7B2%7D+-%5Cfrac%7B1%7D%7B3%7D+-6%29%3D%5C%5C%5C%5C%3D%5Cfrac%7B245%7D%7B2%7D+-+%5Cfrac%7B215%7D%7B3%7D+-30%3D+%5Cfrac%7B125%7D%7B6%7D+%3D20%5Cfrac%7B5%7D%7B6%7D)
Вас заинтересует
4 месяца назад
4 месяца назад
1 год назад
1 год назад
2 года назад
2 года назад